Businessman Donald Trump has a 12-point lead on his nearest 2016 Republican presidential rival in a poll released by Monmouth University on Tuesday — days after another survey also showing Mr. Trump leading in the Granite State was released.

Mr. Trump had the support of 24 percent of likely GOP primary voters in the state, followed by former Florida Gov. Jeb Bush at 12 percent, the Monmouth poll said.

“The controversy over comments about John McCain’s war service do not appear to have slowed the Trump steamroller,” said Patrick Murray, director of the independent Monmouth University Polling Institute in West Long Branch, New Jersey.

The poll was conducted entirely after Mr. Trump’s recent comments that Mr. McCain is a war hero because “he was captured.” Amid criticism, Mr. Trump later clarified multiple times he does believe Mr. McCain is a war hero but did not apologize for the comment, saying Mr. McCain should be the one to apologize for calling supporters who attended a recent Trump rally in Arizona “crazies.”

An NBC-Marist survey released Sunday had Mr. Trump in the lead in New Hampshire with 21 percent, followed by Mr. Bush at 14 percent and Mr. Walker at 12 percent.

In the Monmouth poll, Mr. Trump performed better among registered independents and new voters (29 percent) than among already-registered Republicans (21 percent). But he had a 26-point edge on his nearest competitor among very conservative voters (36 percent to Mr. Walker’s 10 percent) and a 22-point, 35 percent to 13 percent edge over Mr. Walker, the next closest candidate, among tea party supporters.

Mr. Trump also won 30 percent of voters under 50 years old, followed by Mr. Paul’s 12 percent and Mr. Bush’s 10 percent.

Forty-seven percent of primary voters said they have a favorable view of Mr. Trump, compared to 44 percent with an unfavorable one.

“John Kasich says he got into this race because he feels New Hampshire is winnable. It’s too early to assess that likelihood, but he certainly has made an impact there since his announcement,” Mr. Murray said.

The survey of 467 likely GOP primary voters was conducted July 23-26 and has a margin of error of plus or minus 4.5 percent.